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If x and y are positive, is x < y? 13 Feb 2022, 20:07
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Bunuel
If x and y are positive, is x < y?

(1) \(\sqrt{x} < \sqrt{y}\). Since both sides of the inequality are positive (the square root from a positive number is positive), then we can safely square: x < y. Directly answers the question. Sufficient.

(2) \((x-3)^2 < (y-3)^2\). If \(x=3\) and \(y\neq 3\), the inequality will hold true: the left hand side will be 0, while the right hand side will be more than 0. Thus, if \(x=3\), y can be less than 3, giving a NO answer to the question, as well as more than 3, giving an YES answer to the question. Not sufficient.

Answer: A.

Hope it's clear.
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Hey Bunuel. Just had one question. They do not mention anywhere that x is not equal to y. Isn't that a possibility as well ?
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If x and y are positive, is x < y? 14 Feb 2022, 01:38
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Bunuel
If x and y are positive, is x < y?

(1) \(\sqrt{x} < \sqrt{y}\). Since both sides of the inequality are positive (the square root from a positive number is positive), then we can safely square: x < y. Directly answers the question. Sufficient.

(2) \((x-3)^2 < (y-3)^2\). If \(x=3\) and \(y\neq 3\), the inequality will hold true: the left hand side will be 0, while the right hand side will be more than 0. Thus, if \(x=3\), y can be less than 3, giving a NO answer to the question, as well as more than 3, giving an YES answer to the question. Not sufficient.

Answer: A.

Hope it's clear.
Hey Bunuel. Just had one question. They do not mention anywhere that x is not equal to y. Isn't that a possibility as well ?
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x = y does not satisfy any of the statements, so this case is not valid.
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If x and y are positive, is x < y? 14 Feb 2022, 01:44
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If x and y are positive, is x < y?

(1) \(\sqrt{x} < \sqrt{y}\). Since both sides of the inequality are positive (the square root from a positive number is positive), then we can safely square: x < y. Directly answers the question. Sufficient.

(2) \((x-3)^2 < (y-3)^2\). If \(x=3\) and \(y\neq 3\), the inequality will hold true: the left hand side will be 0, while the right hand side will be more than 0. Thus, if \(x=3\), y can be less than 3, giving a NO answer to the question, as well as more than 3, giving an YES answer to the question. Not sufficient.

Answer: A.

Hope it's clear.
Hey Bunuel. Just had one question. They do not mention anywhere that x is not equal to y. Isn't that a possibility as well ?

x = y does not satisfy any of the statements, so this case is not valid.
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Okay got it. Thanks a lot !!!
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If x and y are positive, is x < y? 13 Jan 2024, 20:34
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Lettherebelight
If x and y are positive, is x < y?

(1) \(\sqrt{x} < \sqrt{y}\)
(2) \((x-3)^2 < (y-3)^2\)
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Hi KarishmaB MartyMurray chetan2u
I have attached my solution on paper. For statement 2, is it correct to stop where I figure out it could either be x+y <6 & x>y or x+y >6 & x<y. So, I have two possibilities of relationship between x and y. Hence insufficient. Thanks for your feedback on my solution.
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If x and y are positive, is x < y? 13 Jan 2024, 20:51
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Lettherebelight
If x and y are positive, is x < y?

(1) \(\sqrt{x} < \sqrt{y}\)
(2) \((x-3)^2 < (y-3)^2\)

Hi KarishmaB MartyMurray chetan2u
I have attached my solution on paper. For statement 2, is it correct to stop where I figure out it could either be x+y <6 & x>y or x+y >6 & x<y. So, I have two possibilities of relationship between x and y. Hence insufficient. Thanks for your feedback on my solution.
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Yes, in this question, you could stop at the point you have two different solutions as there are no restrictions on x and y except they are positive integers.
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If x and y are positive, is x < y? 14 Jan 2024, 03:17
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Lettherebelight
If x and y are positive, is x < y?

(1) \(\sqrt{x} < \sqrt{y}\)
(2) \((x-3)^2 < (y-3)^2\)
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Use the number line approach for stmnt 2 to avoid writing anything.


(1) \(\sqrt{x} < \sqrt{y}\)
Squaring both sides because principal square roots are always positive, we get
x < y
Sufficient.


(2) \((x-3)^2 < (y-3)^2\)
Since both sides are squares so positive, we can take the square root to get

|x - 3| < |y - 3|
(Check this post if you are not sure how: https://anaprep.com/algebra-squares-and-square-roots/ )

Distance of x from 3 < Distance of y from 3

So imagine the number line. Either is possible.

______ 1 (= y)_________ 2 (=x) _________ 3 ________ 4 (=x) _______ 5 (= y) _______


x has a smaller range around 3 than y, but in some cases x < y (e.g. x = 2 and y = 5) while in other cases y < x (e.g. x = 2, y = 1)

Not Sufficient

Answer (A)

Check: How to use number line on GMAT: https://youtu.be/3gxVx3Y9xJA
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If x and y are positive, is x < y? 17 Jun 2025, 15:08
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